Some New Inequalities With Proofs and Comments on Applications

This article proves several new inequalities. The proved inequalities are all integrated with the floor function and one of them gives a bound estimation for the Euler’s totient of the semiprimes. Detail mathematical deductions are presented and applicable cases for each inequality are also given with technical comments.


Introduction
The importance of the inequalities and their applications are widely known by researchers of science and technology.Among the thousands of inequalities, those that incorporate the floor function are of very special individuality because of their discrete traits and their wide applications in computer science and various technological aspects.Discovering and proving such inequalities always accompany with mathematical skills and smartness, as seen in chapter 3 of Graham's book (Graham, 1994), in chapter 2 of KUANG's book (Jichang Kuang,2010) and in WANG's article (Xingbo WANG,2017).A recent study came across several new such inequalities.This article introduces , proves them and makes comments on their applications.

Preliminaries
This section lists notations, symbols and lemmas that are adopted in this article.

Definition 1
The floor function of a real x, denoted by ⌊x⌋, is an integer that satisfies inequality ⌊x⌋ ≤ x < ⌊x⌋ + 1, or equivalently, ⌊x⌋ ≤ x < ⌊x⌋ + 1.The fraction part x − ⌊x⌋ is denoted by {x}.

Symbols Symbol A =⇒ B means A can derive out B or B is obtained from A.
Lemma 1 For arbitrary positive real numbers x and y, it holds x+y 2 ≥ √ xy, where the equal sign '=' holds if and only Lemma 2 (See in Xingbo WANG,2017) The floor function ⌊x⌋ holds the following properties (P8) and (P12).

Main Results and Proofs
Theorem 1 For arbitrary odd integer n ≥ 7, it holds Proof.Direct calculation shows that, inequality (3) hold for n = 7, 9, 11, 13 because 1 which is contradictory to Lemma 3. Thus when n > 13 the inequality (3) holds.Consequently the theorem holds.
Corollary 1 For arbitrary positive real numbers α,x and y with x > y , it holds Proof.There are two cases to investigate.One is that |x − y| = n is a positive integer, and the other is not.For the first case, ⌊α(x − y)⌋ = ⌊αn⌋ ≤ αn and by Lemma 2 (P12) α ⌊y − x⌋ = −αn ; then it yields ⌊α(x − y)⌋ + α ⌊y − x⌋ ≤ αn − αn = 0.For the second case, x − y is not an integer, then by Theorem 2 and Lemma 2(P12) leftmost nodes of the tree, it will cost quite a lot of time when p is relatively big.Fortunately, WANG's article proves that, there is not a multiple-node before level 1 + ⌊ log 2 p ⌋ and the congruence ( 16) indicates it might have 2 ≡ −1(modp).This time, Theorem 1 shows 1 + ⌊ log 2 p ⌋ < p−1 2 , providing a mathematical foundation to reduce the searching distance.
Actually, Theorem 2 can have further more applicable occasions.Readers can see them in future works.

Application of Theorem 3
In cryptography, factoring a semiprime, especially a big semiprime, say a RSA number, means a successful step towards solving the difficult problem of integer factorization.There are a lot of literatures mentioning the topic.Among the published methods, the one that calculates or guesses the Euler's totient demonstrates particular individuality for its elementary traits, which is easily understood and relatively faster, as were stated in chapter 6.4 of YAN's book (Yan ,2008).Scholars developed several approaches to estimate the bound of ϕ(n), as Kloster (Kloster ,2010), Jie Fang (Jie Fang & Chenglian Liu,2018) and Kurzweg U H(Kurzweg .2012)did.
On reading Jie Fang's article, it is easy to find some errors in its mathematical deduction.For example, in proving Theorem 1 of the article, it alleged p + q ≥ 2 √ n under the assumption n = pq.Actutally, it is wrong because p q leads to that the equal sign '=' does not hold.This error directly results in a wrong upper bound of ϕ(n) in the article.Theorem 3 might provide a thought to its correction.